1. Field of the Invention
The present invention relates to the transmission of light. In particular, the present invention relates to the transformation of laser beams having a Gaussian energy distribution upon their generation to beams having a Bessel energy distribution for transmission.
2. Discussion of Background
Because of diffraction effects, a laser beam diverges as it travels away from the source, its energy decreasing approximately as the square of distance from the source.
The energy distribution of a laser beam has, immediately upon generation, the familiar Gaussian form shown in FIG. 1a. Energy 10 is proportional to e.sup.-x.spsp.2, where x is the perpendicular distance from the beam axis, and inversely proportional to the square of the distance from the source. The effective scale of x grows in proportion to this distance, i.e. the beam diverges and thereby grows weaker with distance. As illustrated schematically in FIG. 1b, laser beam 12 from source 14 has peak energy 16 and half-power width 18. Peak energy 16 decreases and half-power width 18 increases with increasing distance from source 14.
Huygens' Principle states that for any beam of energy propagating as waves, the beam after passage through any plane is indistinguishable from a beam which would be generated by an array of point sources distributed in that plane, each source releasing energy with the amplitude and phase possessed by the original beam when passing through the same point. Energy radiated in the direction of the original beam is in phase and self-reinforcing; energy radiated in other directions is randomly phased and cancels out. This principle provides a simple means of predicting the overall properties of a beam after passage, for example, through a partially-opaque mask.
A beam is termed "self-replicating" if the energy distribution generated by this interaction is the same, or very similar, at any arbitrary downstream points as in the sampling plane. Gaussian beam 12, which diverges and weakens with distance, is thus not self-replicating under Huygens' Principle.
A laser beam, diverging typically at an angle of about 1 milliradian (0.06.degree.), expands by a factor of 10 or more over a distance of 100 meters. The smaller the starting width of the beam, the more serious the diffraction effects and the greater the divergence. Optical techniques can be used to increase the effective starting width of the beam and thereby minimize its divergence. However, these techniques are also diffraction-limited since the beam still has the basic Gaussian energy distribution.
The Gaussian distribution shown in FIG. 1a may be though of as the sum of all possible Bessel functions, much as Fourier analysis shows any periodic waveform to be a sum of sine waves. The Bessel functions are given by: EQU J.sub.n (z)={2.sup.n+1 z.sup.n n!]/[.pi.(2n)!]}.intg..sub.0.sup..pi./2 cos (x sin .phi.)(cos .phi.).sup.2n d.phi.,
where n is the order of a specific Bessel function and may be any integer from zero to infinity. The zero-order Bessel function J.sub.0 represents the Gaussian central peak. The non-zero-order functions J.sub.1, J.sub.2, and so forth represent ring-shaped areas successively further from the center, containing successively smaller fractions of the total energy in the original Gaussian distribution as n increases. The energy density contributed by each Bessel function J.sub.n (z) is proportional to J.sub.n.sup.2 (z). Energy densities for the first three Bessel functions are shown in FIG. 2. It should be understood that each Bessel function is responsible for producing not one, but an infinite number of annular regions, with the innermost normally being the most intense.
Passing a laser beam through an annular, ring-shaped opening centered on the beam suppresses the zero-order Bessel energy distribution and most non-zero distributions as well, leaving one such non-zero distribution dominant; the result may be termed a "Bessel beam." Such a beam is more nearly self-replicating than a Gaussian beam, typically having only one-tenth the divergence of the Gaussian beam from which it was generated. However, in converting a Gaussian to such a Bessel beam, most of the incident light of the Gaussian beam, including the central peak where the intensity is highest, is blocked by the opaque area on either side of the light-transmitting annulus. As a result, the annulus transfers only a small fraction of the original Gaussian beam energy into the emergent Bessel beam. Therefore, although a laser beam having a Bessel energy distribution has lower divergence than a Gaussian beam, obtaining the Bessel distribution from the just-generated Gaussian beam results in substantial losses before any significant transmission begins.